mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Group.SeparationQuotient

Goal state before:
pp: M : Type u_1
N✝ : Type u_2
inst✝² : SeminormedAddCommGroup M
inst✝¹ : SeminormedAddCommGroup N✝
N : Type u_3
inst✝ : SeminormedAddCommGroup N
g : NormedAddGroupHom (SeparationQuotient M) N
⊢ ∀ (x : M), ‖x‖ = 0 → (g.comp normedMk) x = 0

Tactic:
intro x

Goal state after:
M : Type u_1
N✝ : Type u_2
inst✝² : SeminormedAddCommGroup M
inst✝¹ : SeminormedAddCommGroup N✝
N : Type u_3
inst✝ : SeminormedAddCommGroup N
g : NormedAddGroupHom (SeparationQuotient M) N
x : M
⊢ ‖x‖ = 0 → (g.comp normedMk) x = 0
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsUltrametricDist K
h_alg : Algebra.IsAlgebraic K L
⊢ IsNonarchimedean (spectralNorm K L)

Tactic:
intro x

Goal state after:
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsUltrametricDist K
h_alg : Algebra.IsAlgebraic K L
x : L
⊢ ∀ (b : L), spectralNorm K L (x + b) ≤ max (spectralNorm K L x) (spectralNorm K L b)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
⊢ ∀ (y : L),
    ∃ C₁ C₂,
      ∃ (_ : 0 < C₁) (_ : 0 < C₂),
        ∀ (x : ↥K[y]), f ↑x ≤ C₁ * (spectralAlgNorm K L) ↑x ∧ (spectralAlgNorm K L) ↑x ≤ C₂ * f ↑x

Tactic:
intro x

Goal state after:
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
x : L
⊢ ∃ C₁ C₂,
    ∃ (_ : 0 < C₁) (_ : 0 < C₂),
      ∀ (x_3 : ↥K[x]), f ↑x_3 ≤ C₁ * (spectralAlgNorm K L) ↑x_3 ∧ (spectralAlgNorm K L) ↑x_3 ≤ C₂ * f ↑x_3
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
⊢ IsPowMul (smoothingFun μ)

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
⊢ ∀ {n : ℕ}, 1 ≤ n → smoothingFun μ (x ^ n) = smoothingFun μ x ^ n
Module: Mathlib.Analysis.Normed.Group.Tannery

Goal state before:
pp: case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
⊢ ∀ ε > 0, ∀ᶠ (x : α) in 𝓕, dist (∑' (k : β), f x k) (∑' (k : β), g k) < ε

Tactic:
intro ε hε

Goal state after:
case inr.inr
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
h_suma : ∀ᶠ (n : α) in 𝓕, Summable fun x ↦ ‖f n x‖
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x : α) in 𝓕, dist (∑' (k : β), f x k) (∑' (k : β), g k) < ε
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x y : L
⊢ spectralNorm K L (x - y) = spectralNorm K L (y - x)

Tactic:
← neg_sub,

Goal state after:
R : Type u_1
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x y : L
⊢ spectralNorm K L (-(y - x)) = spectralNorm K L (y - x)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
⊢ ((mapAlg K E) (minpoly K x)).natDegree = ((mapAlg K E) (minpoly K x)).roots.card

Tactic:
· rw [eq_comm, ← splits_iff_card_roots]
      exact IsSplittingField.IsScalarTower.splits (K := L) E (minpoly K x)

Goal state after:
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: case h
R : Type u_1
inst✝⁶ : Semiring R
F : Type u_4
G : Type u_6
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
⊢ LeftInverse ⇑(fst R F G) ⇑(inl R F G)

Tactic:
intro x

Goal state after:
case h
R : Type u_1
inst✝⁶ : Semiring R
F : Type u_4
G : Type u_6
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
x : F
⊢ (fst R F G) ((inl R F G) x) = x
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: case h
R : Type u_1
inst✝⁶ : Semiring R
F : Type u_4
G : Type u_6
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
⊢ LeftInverse ⇑(snd R F G) ⇑(inr R F G)

Tactic:
intro x

Goal state after:
case h
R : Type u_1
inst✝⁶ : Semiring R
F : Type u_4
G : Type u_6
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
x : G
⊢ (snd R F G) ((inr R F G) x) = x
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: case h
R : Type u_1
inst✝⁶ : Semiring R
F : Type u_4
G : Type u_6
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
⊢ RightInverse ⇑((ContinuousLinearMap.id R F).prod 0) ⇑(fst R F G)

Tactic:
intro x

Goal state after:
case h
R : Type u_1
inst✝⁶ : Semiring R
F : Type u_4
G : Type u_6
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
x : F
⊢ (fst R F G) (((ContinuousLinearMap.id R F).prod 0) x) = x
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: case h
R : Type u_1
inst✝⁶ : Semiring R
F : Type u_4
G : Type u_6
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
⊢ RightInverse ⇑(ContinuousLinearMap.prod 0 (ContinuousLinearMap.id R G)) ⇑(snd R F G)

Tactic:
intro x

Goal state after:
case h
R : Type u_1
inst✝⁶ : Semiring R
F : Type u_4
G : Type u_6
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
x : G
⊢ (snd R F G) ((ContinuousLinearMap.prod 0 (ContinuousLinearMap.id R G)) x) = x
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q : M
h₁ : IsLprojection X P
h₂ : IsLprojection X Q
⊢ ∀ (x : X), ‖x‖ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖

Tactic:
intro x

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q : M
h₁ : IsLprojection X P
h₂ : IsLprojection X Q
x : X
⊢ ‖x‖ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: case refine_2
ι : Type u_1
R : Type u_2
inst✝² : NormedCommRing R
inst✝¹ : NormOneClass R
f : ι → R
inst✝ : CompleteSpace R
hf : Summable fun i ↦ ‖f i‖
ε : ℝ
hε : ε > 0
r₁ : ℝ
hr₁ : r₁ > 0
s₁ : Finset ι
hs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁
s₂ : Finset ι
hs₂ : ∀ (t : Finset ι), Disjoint t s₂ → ‖∏ i ∈ t, (1 + f i) - 1‖ < ε / (2 * r₁)
s : Finset ι := s₁ ∪ s₂
⊢ ∀ x ∈ Metric.ball (∏ i ∈ s, (1 + f i)) (ε / 2), ∀ y ∈ Metric.ball (∏ i ∈ s, (1 + f i)) (ε / 2), dist x y < ε

Tactic:
intro x

Goal state after:
case refine_2
ι : Type u_1
R : Type u_2
inst✝² : NormedCommRing R
inst✝¹ : NormOneClass R
f : ι → R
inst✝ : CompleteSpace R
hf : Summable fun i ↦ ‖f i‖
ε : ℝ
hε : ε > 0
r₁ : ℝ
hr₁ : r₁ > 0
s₁ : Finset ι
hs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁
s₂ : Finset ι
hs₂ : ∀ (t : Finset ι), Disjoint t s₂ → ‖∏ i ∈ t, (1 + f i) - 1‖ < ε / (2 * r₁)
s : Finset ι := s₁ ∪ s₂
x : R
⊢ x ∈ Metric.ball (∏ i ∈ s, (1 + f i)) (ε / 2) → ∀ y ∈ Metric.ball (∏ i ∈ s, (1 + f i)) (ε / 2), dist x y < ε
Module: Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
𝕜 : Type u_2
E : ι → Type u_3
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : NormedField 𝕜
p : FreeAddMonoid (𝕜 × ((i : ι) → E i))
a : ℝ
⊢ ∀ (x : 𝕜) (x_1 : (i : ι) → E i), (x, x_1) ∈ FreeAddMonoid.toList p → ‖x‖ * ∏ x, ‖x_1 x‖ = a → 0 ≤ a

Tactic:
intro x

Goal state after:
ι : Type u_1
inst✝² : Fintype ι
𝕜 : Type u_2
E : ι → Type u_3
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : NormedField 𝕜
p : FreeAddMonoid (𝕜 × ((i : ι) → E i))
a : ℝ
x : 𝕜
⊢ ∀ (x_1 : (i : ι) → E i), (x, x_1) ∈ FreeAddMonoid.toList p → ‖x‖ * ∏ x, ‖x_1 x‖ = a → 0 ≤ a
Module: Mathlib.Analysis.Normed.Lp.lpHolder

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : ι → Type u_3
F : ι → Type u_4
G : ι → Type u_5
inst✝⁶ : RCLike 𝕜
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : (i : ι) → NormedAddCommGroup (F i)
inst✝² : (i : ι) → NormedSpace 𝕜 (F i)
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → NormedSpace 𝕜 (G i)
p q r : ℝ≥0∞
hpqr✝ : p.HolderTriple q r
e : (i : ι) → E i
f : (i : ι) → F i
hp : 0 < p.toReal
hq : 0 < q.toReal
B : (i : ι) → E i →L[𝕜] F i →L[𝕜] G i
K C D : ℝ
hBK : ∀ (i : ι), ‖B i‖ ≤ K
hK : 0 ≤ K
hC : 0 ≤ C
hCe : ∀ (s : Finset ι), ∑ i ∈ s, ‖e i‖ ^ p.toReal ≤ C
hDf : ∀ (s : Finset ι), ∑ i ∈ s, ‖f i‖ ^ q.toReal ≤ D
s : Finset ι
hpqr : p.toReal.HolderTriple q.toReal r.toReal
hr : 0 < r.toReal
this : ∑ i ∈ s, (‖e i‖ * ‖f i‖) ^ r.toReal ≤ C ^ (r.toReal / p.toReal) * D ^ (r.toReal / q.toReal)
⊢ K ^ r.toReal * ∑ i ∈ s, (‖e i‖ * ‖f i‖) ^ r.toReal ≤
    K ^ r.toReal * C ^ (r.toReal / p.toReal) * D ^ (r.toReal / q.toReal)

Tactic:
rw [mul_assoc]
      gcongr

Goal state after:
Module: Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm

Goal state before:
pp: case refine_2
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
w✝¹ : SeminormedAddCommGroup G
w✝ : NormedSpace 𝕜 G
h : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
⊢ (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G) ≤ projectiveSeminorm

Tactic:
intro x

Goal state after:
case refine_2
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
w✝¹ : SeminormedAddCommGroup G
w✝ : NormedSpace 𝕜 G
h : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
x : ⨂[𝕜] (i : ι), E i
⊢ ((normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)) x ≤
    projectiveSeminorm x
Module: Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁶] ↑A
inst✝ : A.CoFG
h_ker : Disjoint (↑u).ker A
F' : Submodule 𝕜 F := (↑u).range
h_range : IsClosed[inst✝³] ↑F'
i : ↥F' →L[𝕜] F := F'.subtypeL
i_clemb : IsClosedEmbedding ⇑i
u' : E →L[𝕜] ↥F' := u.rangeRestrict
range_u' : (↑u').range = ⊤
eq1 : u = i ∘SL u'
eq2 : u.domRestrict A = i ∘SL u'.domRestrict A
⊢ IsStrictMap ⇑u ↔ IsStrictMap ⇑(u.domRestrict A) ∧ IsClosed[inst✝³] ↑(↑(u.domRestrict A)).range

Tactic:
eq2,

Goal state after:
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁶] ↑A
inst✝ : A.CoFG
h_ker : Disjoint (↑u).ker A
F' : Submodule 𝕜 F := (↑u).range
h_range : IsClosed[inst✝³] ↑F'
i : ↥F' →L[𝕜] F := F'.subtypeL
i_clemb : IsClosedEmbedding ⇑i
u' : E →L[𝕜] ↥F' := u.rangeRestrict
range_u' : (↑u').range = ⊤
eq1 : u = i ∘SL u'
eq2 : u.domRestrict A = i ∘SL u'.domRestrict A
⊢ IsStrictMap ⇑u ↔ IsStrictMap ⇑(i ∘SL u'.domRestrict A) ∧ IsClosed[inst✝³] ↑(↑(i ∘SL u'.domRestrict A)).range
Module: Mathlib.Analysis.Normed.Operator.Fredholm.Basic

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 E
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
u_fred : u.IsFredholm
dom₁ : Submodule 𝕜 E
codom₀ : Submodule 𝕜 F
h_dom : IsTopCompl (↑u).ker dom₁
h_codom : IsTopCompl (↑u).range codom₀
⊢ Set.EqOn ⇑u
    ⇑({ X₀ := codom₀, X₁ := (↑u).range, isTopCompl := h_codom, finite_X₀ := ⋯ }.X₁.subtypeL ∘SL
        ↑(((↑u).ker.quotientEquivOfIsTopCompl dom₁ h_dom).symm.trans (ContinuousLinearEquiv.quotKerEquivRange ⋯)) ∘SL
          { X₀ := (↑u).ker, X₁ := dom₁, isTopCompl := ⋯, finite_X₀ := ⋯ }.proj)
    ↑(↑u).ker

Tactic:
intro x

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 E
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
u_fred : u.IsFredholm
dom₁ : Submodule 𝕜 E
codom₀ : Submodule 𝕜 F
h_dom : IsTopCompl (↑u).ker dom₁
h_codom : IsTopCompl (↑u).range codom₀
x : E
⊢ x ∈ ↑(↑u).ker →
    u x =
      ({ X₀ := codom₀, X₁ := (↑u).range, isTopCompl := h_codom, finite_X₀ := ⋯ }.X₁.subtypeL ∘SL
          ↑(((↑u).ker.quotientEquivOfIsTopCompl dom₁ h_dom).symm.trans (ContinuousLinearEquiv.quotKerEquivRange ⋯)) ∘SL
            { X₀ := (↑u).ker, X₁ := dom₁, isTopCompl := ⋯, finite_X₀ := ⋯ }.proj)
        x
Module: Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁶] ↑A
inst✝ : A.CoFG
N : Submodule 𝕜 E := A ⊓ (↑u).ker
π : E →L[𝕜] E ⧸ N := N.mkQL
v : E ⧸ N →L[𝕜] F := N.liftQL u ⋯
π_quot : IsOpenQuotientMap ⇑π
u_eq : u = v ∘SL π
B : Submodule 𝕜 (E ⧸ N) := map N.mkQ A
B_cofg : B.CoFG
⊢ Disjoint (↑v).ker B

Tactic:
simp [disjoint_iff, v, B, toLinearMap_liftQL, ker_liftQ,
      map_inf_eq_map_inf_comap, comap_map_mkQ, N, inf_comm]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Fredholm.Basic

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 E
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
u_fred : u.IsFredholm
dom₁ : Submodule 𝕜 E
codom₀ : Submodule 𝕜 F
h_dom : IsTopCompl (↑u).ker dom₁
h_codom : IsTopCompl (↑u).range codom₀
⊢ Set.EqOn ⇑u
    ⇑({ X₀ := codom₀, X₁ := (↑u).range, isTopCompl := h_codom, finite_X₀ := ⋯ }.X₁.subtypeL ∘SL
        ↑(((↑u).ker.quotientEquivOfIsTopCompl dom₁ h_dom).symm.trans (ContinuousLinearEquiv.quotKerEquivRange ⋯)) ∘SL
          { X₀ := (↑u).ker, X₁ := dom₁, isTopCompl := ⋯, finite_X₀ := ⋯ }.proj)
    ↑dom₁

Tactic:
intro x

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 E
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : CompleteSpace 𝕜
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
u_fred : u.IsFredholm
dom₁ : Submodule 𝕜 E
codom₀ : Submodule 𝕜 F
h_dom : IsTopCompl (↑u).ker dom₁
h_codom : IsTopCompl (↑u).range codom₀
x : E
⊢ x ∈ ↑dom₁ →
    u x =
      ({ X₀ := codom₀, X₁ := (↑u).range, isTopCompl := h_codom, finite_X₀ := ⋯ }.X₁.subtypeL ∘SL
          ↑(((↑u).ker.quotientEquivOfIsTopCompl dom₁ h_dom).symm.trans (ContinuousLinearEquiv.quotKerEquivRange ⋯)) ∘SL
            { X₀ := (↑u).ker, X₁ := dom₁, isTopCompl := ⋯, finite_X₀ := ⋯ }.proj)
        x
Module: Mathlib.Analysis.Normed.Order.UpperLower

Goal state before:
pp: case intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
hs : IsClosed s
hs' : BddBelow s
val✝ : Fintype ι
f : ℕ → ι → ℝ
x : ι → ℝ
hx : Filter.Tendsto f Filter.atTop (nhds x)
g : ℕ → ι → ℝ
hg : ∀ (n : ℕ), g n ∈ s
hgf : ∀ (n : ℕ), g n ≤ f n
⊢ x ∈ ↑(upperClosure s)

Tactic:
obtain ⟨a, ha⟩ := hx.bddAbove_range

Goal state after:
case intro
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
hs : IsClosed s
hs' : BddBelow s
val✝ : Fintype ι
f : ℕ → ι → ℝ
x : ι → ℝ
hx : Filter.Tendsto f Filter.atTop (nhds x)
g : ℕ → ι → ℝ
hg : ∀ (n : ℕ), g n ∈ s
hgf : ∀ (n : ℕ), g n ≤ f n
a : ι → ℝ
ha : a ∈ upperBounds (range f)
⊢ x ∈ ↑(upperClosure s)
Module: Mathlib.Topology.MetricSpace.Contracting

Goal state before:
pp: α : Type u_1
inst✝ : EMetricSpace α
K : ℝ≥0
f : α → α
hf : ContractingWith K f
x y : α
hx : IsFixedPt f x
hy : IsFixedPt f y
⊢ x = y ∨ edist x y = ∞

Tactic:
refine or_iff_not_imp_right.2 fun h ↦ edist_le_zero.1 ?_
  simpa only [hx.eq, edist_self, add_zero, ENNReal.zero_div] using hf.edist_le_of_fixedPoint h hy

Goal state after:
Module: Mathlib.Analysis.ODE.Gronwall

Goal state before:
pp: f f' : ℝ → ℝ
δ K ε a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r
ha : f a ≤ δ
bound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε
⊢ ∀ x ∈ Icc a b, ∀ ε' ∈ Ioi ε, f x ≤ gronwallBound δ K ε' (x - a)

Tactic:
intro x

Goal state after:
f f' : ℝ → ℝ
δ K ε a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r
ha : f a ≤ δ
bound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε
x : ℝ
⊢ x ∈ Icc a b → ∀ ε' ∈ Ioi ε, f x ≤ gronwallBound δ K ε' (x - a)
Module: Mathlib.Analysis.ODE.Gronwall

Goal state before:
pp: f f' : ℝ → ℝ
δ K ε a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r
ha : f a ≤ δ
bound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε
H : ∀ x ∈ Icc a b, ∀ ε' ∈ Ioi ε, f x ≤ gronwallBound δ K ε' (x - a)
⊢ ∀ x ∈ Icc a b, f x ≤ gronwallBound δ K ε (x - a)

Tactic:
intro x

Goal state after:
f f' : ℝ → ℝ
δ K ε a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r
ha : f a ≤ δ
bound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε
H : ∀ x ∈ Icc a b, ∀ ε' ∈ Ioi ε, f x ≤ gronwallBound δ K ε' (x - a)
x : ℝ
⊢ x ∈ Icc a b → f x ≤ gronwallBound δ K ε (x - a)
Module: Mathlib.Analysis.ODE.Gronwall

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
K a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
ha : f a = 0
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖
⊢ ∀ x ∈ Icc a b, f x = 0

Tactic:
intro x

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
K a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
ha : f a = 0
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖
x : ℝ
⊢ x ∈ Icc a b → f x = 0
Module: Mathlib.Analysis.ODE.ExistUnique

Goal state before:
pp: case refine_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
α : (x : E) → x ∈ closedBall x₀ ↑r → FunSpace t₀ x₀ r L
hα : ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), IsFixedPt (FunSpace.next hf hx) (α x hx)
α' : E → ℝ → E := fun x ↦ if hx : x ∈ closedBall x₀ ↑r then (α x hx).compProj else 0
hα' : α' = fun x ↦ if hx : x ∈ closedBall x₀ ↑r then (α x hx).compProj else 0
L' : ℝ≥0
h :
  ∀ (x y : E) (hx : x ∈ closedBall x₀ ↑r) (hy : y ∈ closedBall x₀ ↑r) (α β : FunSpace t₀ x₀ r L),
    IsFixedPt (FunSpace.next hf hx) α → IsFixedPt (FunSpace.next hf hy) β → dist α β ≤ ↑L' * dist x y
t : ℝ
ht : t ∈ Icc tmin tmax
x : E
hx : x ∈ closedBall x₀ ↑r
y : E
hy : y ∈ closedBall x₀ ↑r
this : Nonempty ↑(Icc tmin tmax)
⊢ dist ((FunSpace.toContinuousMap (α x hx)) (projIcc tmin tmax ⋯ t))
      ((FunSpace.toContinuousMap (α y hy)) (projIcc tmin tmax ⋯ t)) ≤
    ↑L' * dist x y

Tactic:
apply ContinuousMap.dist_le_iff_of_nonempty.mp

Goal state after:
case refine_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
α : (x : E) → x ∈ closedBall x₀ ↑r → FunSpace t₀ x₀ r L
hα : ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), IsFixedPt (FunSpace.next hf hx) (α x hx)
α' : E → ℝ → E := ⋯
hα' : α' = fun x ↦ if hx : x ∈ closedBall x₀ ↑r then (α x hx).compProj else 0
L' : ℝ≥0
h :
  ∀ (x y : E) (hx : x ∈ closedBall x₀ ↑r) (hy : y ∈ closedBall x₀ ↑r) (α β : FunSpace t₀ x₀ r L),
    IsFixedPt (FunSpace.next hf hx) α → IsFixedPt (FunSpace.next hf hy) β → dist α β ≤ ↑L' * dist x y
t : ℝ
ht : t ∈ Icc tmin tmax
x : E
hx : x ∈ closedBall x₀ ↑r
y : E
hy : y ∈ closedBall x₀ ↑r
this : Nonempty ↑(Icc tmin tmax)
⊢ dist (FunSpace.toContinuousMap (α x hx)) (FunSpace.toContinuousMap (α y hy)) ≤ ↑L' * dist x y